Fixed subgroups and computation of auto-fixed closures in free-abelian\n times free groups

2019/06/05 by Mallika Roy, Enric Ventura, Roy, Mallika +1 · 2 citations
Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1906.02144

openalex publication_date 2019/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The classical result by Dyer--Scott about fixed subgroups of finite order\nautomorphisms of F_n being free factors of F_n is no longer true in\nZ^m\× F_n. Within this more general context, we prove a relaxed version\nin the spirit of Bestvina--Handel Theorem: the rank of fixed subgroups of\nfinite order automorphisms is uniformly bounded in terms of m,n. We also\nstudy periodic points of endomorphisms of Z^m\× F_n, and give an\nalgorithm to compute auto-fixed closures of finitely generated subgroups of\nZ^m\× F_n. On the way, we prove the analog of Day's Theorem for real\nelements in Z^m\× F_n, contributing a modest step into the project of\ndoing so for any right angled Artin group (as McCool did with respect to\nWhitehead's Theorem in the free context).\n

Cited by

• On the finiteness properties of fixed subgroups of automorphisms
• Fixed subgroups of generalised Baumslag-Solitar groups

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• Computation of endo-fixed closures in free-abelian times free groups
• A description of auto-fixed subgroups in a free group
• Algorithmic problems for free-abelian times free groups
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• Computing fixed closures in free groups